find the limit of the following sequence or determine that the sequence diverges. {3n^2 - 1 / 2n^2 + 1}…
find the limit of the following sequence or determine that the sequence diverges. {3n^2 - 1 / 2n^2 + 1} select the correct choice below and fill in any answer boxes to complete the choice. a. the limit of the sequence is. (type an exact answer.) b. the sequence diverges.
Answer
Explanation:
Step1: Divide numerator and denominator by $n^{2}$
$\lim_{n\rightarrow\infty}\frac{3n^{2}-1}{2n^{2}+1}=\lim_{n\rightarrow\infty}\frac{3-\frac{1}{n^{2}}}{2 + \frac{1}{n^{2}}}$
Step2: Evaluate the limit
As $n\rightarrow\infty$, $\frac{1}{n^{2}}\rightarrow0$. So $\lim_{n\rightarrow\infty}\frac{3-\frac{1}{n^{2}}}{2+\frac{1}{n^{2}}}=\frac{3 - 0}{2+0}=\frac{3}{2}$
Answer:
A. The limit of the sequence is $\frac{3}{2}$